Page 110 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 97

Figure 2.13 Global shape functions.

N
T=~T' =N~~+.-+N,~+.-N,T~ =~N~T~
(2.72)
j=l
This completes the specification of the basis functions. Now we can return to the
variational formulation again.

Step3: Assembling the Element Equations to Form the Global Problem
In step 2 we derived the approximation function for T. Galerkin's formulation
assumes the weight function to be same as the approximation for the unknown
variables. Therefore we have w = T. With this, we can substitute T and w in
(2.66).

Ik Ib Is
For clarity we consider terms of (2.73) separately. The equation (2.73) actually
gives the error of the approximate solution (refer to exercise 2.4). To minimise
the error, equation (2.73) should be differentiated w.r.t Ti which are the
coefficients of the polynomials. The minimization process converts Ik into the
stiffness matrix K. For four nodes (i.e. three elements as in figure 2.1 lc) we can
expand Ik as below. In what follows we indicate the limits as xi and xj. This is to
reduce the complications that arising in evaluating the terms. The intervals in the
integrals depend on global shape functions.